1. Field of the Invention
The present invention relates generally to the field of range navigation systems.
2. Description of the Background Art
In the available art related to range navigation systems, GPS and other navigation systems commonly measure the ranges to several transmitters by observing the times at which signals transmitted at earlier known times are received, wherein the distance is generally proportional to the measured time difference. The measurements are commonly related to an assumed position by subtracting from each range measurement a prediction of the range from that assumed position to the satellite or other navigation transmitter. The measurements are also intrinsically related to the time assumed in making the measurements, since any error in that time appears as an offset in the measured ranges. Such systems then commonly obtain a position fix by updating the assumed position and time on the basis of the discrepancies between the range measurements and predictions.
Although the relationship between the measurements and the physical coordinates of the vehicle is generally nonlinear, navigation systems commonly linearize the relationship about the assumed position and time, using the easily-computed first derivatives of the measurements with respect to the physical coordinates. This assumption of linearity allows the measurement-prediction discrepancies to be expressed easily as a function of the unknown error in the assumed position and time, specifically, as a matrix equation: EQU z=hx; (1)
where z is the measurement-discrepancy vector, h is the matrix of partial derivatives of, commonly, three spatial and one temporal physical coordinates, and x is the update vector. The relationship (1) is then easily inverted by well-known matrix methods to express x as a function of z: EQU x=h.sup.-1 z=nz; (2) PA1 where n is the navigation matrix. The resulting vector x in physical coordinates is then commonly applied as an update to the assumed position and time. Furthermore, there are well-known techniques to optimize that update when an excess of measurements over unknowns over-determines the solution, as by the least-squares or weighted-least-squares criteria.
Errors in the elements of z cause errors in the update x. If the z errors are assumed to be uncorrelated and equal in variance, as is generally true of GPS measurements, then the variance of the resulting position-update error is the product of the variance of each element of z with the sum of the squares of the elements of the rows of n that correspond to the three Cartesian position coordinates, by virtue of the statistical orthogonality of the measurements and the geometric orthogonality of the position coordinates. The square-root of that sum of squares is therefore the ratio of the standard deviation of position error to the standard deviation of measurement error and is commonly termed as the "position dilution of precision" or PDOP.
Although this treatment of an essentially nonlinear relationship as linear causes inevitable error in each update, that error is ordinarily negligible, because of the great disparity between the magnitudes of the update on one hand and the distances to the satellites on the other. Moreover, this same consideration makes it unnecessary in many applications to recompute the linearized relationship for each update, since it changes so little. Finally, if the linearized relationship causes update errors to become unacceptable though small compared to the update's size, iteration of the update process with the same set of measurements can reduce the need for frequent recomputation of the update matrix.
Once the measurement-prediction discrepancies z are available, the computation of the update x is a straightforward and relatively inexpensive matrix multiplication. However, the calculation of the predictions of the ranges from the assumed position to the satellites is more difficult, in that it generally requires a square-root operation for each satellite. This follows from the Pythagorean relationship that the square of the range is the sum of the squares of the differences in the three Cartesian coordinates commonly used to express the assumed and satellite positions. Thus, the predicted range to be subtracted from the measured range is the square-root of that sum.
What is needed is an unconventional range navigation system with efficient update process that avoids the frequent recalculation of square-roots in range predictions.